**Search Mathematics of the DFT**

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This chapter provides an introduction to sinusoids, exponentials, complex sinusoids, and various associated terminology, such as exponential decay-time ``'', in-phase and quadrature sinusoidal components, analytic signals, positive and negative frequencies, and constructive and destructive interference. The fundamental importance of sinusoids in the analysis of linear time-invariant systems is introduced. We also look at circular motion expressed as the vector sum of in-phase and quadrature sinusoidal motions. Both continuous and discrete-time sinusoids are considered. In particular, a sampled complex sinusoid is generated by successive powers of any complex number .

- Sinusoids
- Example Sinusoids
- Why Sinusoids are Important
- In-Phase & Quadrature Sinusoidal Components
- Sinusoids at the Same Frequency
- Constructive and Destructive Interference
- Sinusoid Magnitude Spectra

- Exponentials

- Complex Sinusoids
- Circular Motion
- Projection of Circular Motion
- Positive and Negative Frequencies
- Plotting Complex Sinusoids versus Frequency
- Sinusoidal Amplitude Modulation (AM)
- Sinusoidal Frequency Modulation (FM)
- Analytic Signals and Hilbert Transform Filters
- Generalized Complex Sinusoids
- Sampled Sinusoids
- Powers of
*z* - Phasors and Carriers
- Importance of Generalized Complex Sinusoids
- Comparing Analog and Digital Complex Planes

- Sinusoid Problems

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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